We’ve started the second section of Unit 4, learning about complex and imaginary numbers. This is a new category of number, much like irrational or negative numbers, that mathematicians developed to handle equations that they previously had no ability to solve. We watched two videos to introduce this topic, but I recommend you watch the full story found here.

Note: Due to a delay in our completion of Friday’s lesson, this assignment will be postponed for Monday, February 3

We’ve started the second section of Unit 4, learning about complex and imaginary numbers. This is a new category of number, much like irrational or negative numbers, that mathematicians developed to handle equations that they previously had no ability to solve. We watched two videos to introduce this topic, but I recommend you watch the full story found here.

The third Personal Progress Check for Unit 4 has been assigned. Log into AP Classroom and complete the Unit 4 MCQ Part C PPC by the start of class on Tuesday, February 4.

Out Unit 4 Test has been rescheduled for Thursday, February 6. We’ll wrap up Chapter 16 in class on Monday, then review the unit as a whole on Tuesday and Wednesday. Refer to yesterday’s post for the review assignment.

Out Unit 4 Test will be on Wednesday, February 5. We’ll wrap up Chapter 16 in class on Friday, then review the unit on Monday and Tuesday. I recommend you get started on the following practice problems from the Unit 4 review, pages 434-439: 5, 16, 20, 21, 23, 25, 27, 28, 37.

From chapter 16, read pages 418-423, up to the section titled “The Normal Model to the Rescue!”

From the exercises on pages 430-432, do 7, 10, 12, 17

A word about assumptions and conditions…

In order for a model (e.g., a normal model or binomial probability model) to be valid for a scenario, we must be able to make certain assumptions about that scenario. These assumptions can include the results of individual trials being independent from each other, the distribution of results being sufficiently unimodal and symmetric, etc.

If it is appropriate to make these assumptions, then go ahead and do it. But if it is not appropriate to make these assumptions, you can still proceed provided certain conditions are satisfied. These conditions mean that the scenario is “close enough” to allow the model to be valid.

For example, coin flips are independent. There is no finite population of coin flips that you are “drawing” a sample from, and so each coin flip’s outcome is independent of the next. Drawing cards from a deck are not independent, as the deck is finite and the probability of a certain outcome changes with each card that is removed. However, if the population is large enough, or more specifically if the sample is small enough in comparison to the population, then that probability change is very small, small enough to be ignored.

In general, as long as the sample size is less than 10% of the overall population, the probability change isn’t big enough to be worrisome. The reason why the magic number is 10% has to do with something called the Finite Population Correction Factor, and a thorough description of where it comes from and how it affects probabilities can be found here.

With the start of the second semester, we have introduced some new classroom norms. Think of these as general expectations that I have of you and that you should have of each other during group work and class discussions. We did an activity today that would put the norms into practice.

If you are here to submit a course evaluation, please do so here, and thank you. If you enjoyed this class, please mention it to any of your friends who may be considering signing up in the 2020/2021 school year. Word of mouth is the best advertising for the class!

We discussed an open problem related to the connectedness of the Mandelbrot Set today. It seems silly, but “connected” is actually a tough idea to define mathematically. We discussed the following ideas:

Not disconnected: That is, the set is not divided into pieces. Pick any two points that are members of the set, and we can’t draw a line that separates those two points without crossing the set somewhere. The Mandelbrot set is not disconnected

“Simply” connected: Both the set and its complement (everything not a member of the set) are not disconnected. A circle (defined by the equation x² + y² = 25, for example) is not simply connected, because it divides the plane into regions inside and outside of the circle, which are disconnected from each other by the circle. A disk — a circle that includes its interior, for example defined by x² + y² < 25 — is simply connected, as anything not a part of the disk is outside the disk. The Mandelbrot Set is simply connected

“Path” connected:Any two points of the set can be joined by a path that stay in the set. We used the Topologist’s Sine Curve to define an example that is not “path” connected. The Mandelbrot Set, though, is path connected.

“Locally” connected: Around any point of the set, we can draw a circle small enough such that the set is not connected inside of it. We used the idea of a comb defined by the line segments y = 1/2^x from x: [0,1] as an example of a space not locally connected. Mathematicians are still unsure whether the Mandelbrot is Locally Connected or not, an open problem known as the MLC.

Our last meeting will be on Wednesday, January 22nd in room G116. Our last topic will be a a brief history lesson of the discovery of the Julia Set that starts in Newton’s Method for Approximation.

By next Wednesday, please read this article from the November, 1991 issue of Science News, published just after an important discovery was made about the border of the Mandelbrot Set, as well as this follow-up of sorts from 2017 in Scientific American

To review, take a look at the following exercises from pages 408-411: 19, 20, 21, 36, 38, 41, 42, 52, and 55. You can find answers here. Leave a comment or send me an email with questions!

Your next Personal Progress Check has been assigned. Log into AP Classroom and complete the Unit 4 MCQ Part B PPC. Your response is due by the start of class time (9:00 for 1st period, 2:00 for 7th) on Monday, January 20

Update (11/22): I have reopened the PPC so that those of you who did not complete it within the expected timeframe can refer to the questions as review for your quiz on Monday. Completing it now will not be for credit.

We wrapped up yesterday’s game of “Find the Julia Set” and reviewed some other interesting properties of the Mandelbrot Set (many of which were hinted at by Mr. Drix yesterday):

“Area” Julia Sets come from “areas” in the Mandelbrot Set, and “string” Julia Sets come from “strings” in the Mandelbrot Set

There are “veins” through the main cardioid of the Mandelbrot Set where we can watch the orbit of a single fixed point die towards a 2-, 3-, 4-, or any magnitude cycle. Along these veins, the fixed point forms spokes. Offset slightly from these veins and they become spirals

Relatedly, there are balls off of the main cardioid for every possible cycle, and every variant of every cycle (1/5-cycles, 2/5-cycles, etc.). Imagine a circle centered on the cusp of the cardioid. Any angle rotation around that circle points straight at a ball on the edge of the cardioid, whose orbit-cycle magnitude exactly matches the fraction used to find that angle. A 120° rotation, representing 1/3 of a full rotation, points straight at the 3-cycle ball on the top of the Mandelbrot set, and a 240° rotation, 2/3 a full rotation, points straight at the 3-cycle ball on the bottom. A 72° rotation, 1/5 of a full, points straight at the first 5-ball. Rotations of 144°, 216°, or 288° point straight at the second, third, and fourth 5-balls.

This note also explains why we can only find two 6 balls along the edge of the cardioid. A 1/6 (60°) rotation points to a six ball, but a 2/6 rotation points to the 3-ball (2/6 = 1/3), a 3/6 rotation points to the 2-ball (3/6 = 1/2), and a 4/6 rotation points to the other 3-ball (4/6 = 2/3). The only two 6-balls are at 1/6 and 5/6 rotations around the cusp.

Our last two topics of the course: an open problem related to the connectedness of the Mandelbrot Set, and a discussion of Newton’s Method of Approximation, and its surprising connection to the Mandelbrot Set

We will have a quiz on chapter 15 when we return from exam week: Monday, January 27

Your next Personal Progress Check has been assigned. Log into AP Classroom and complete the Unit 4 MCQ Part B PPC. Your response is due by the start of class time (9:00 for 1st period, 2:00 for 7th) on Monday, January 20

The central question we have remaining is: In what way are the shape of the Julia Set, the size of the cycle with it, and its location in the Mandelbrot Set related to each other? We’ve discussed parts of this question, but today with Mr. Drix we answered it in earnest.

First, we observed that orbits along the horizontal x-axis of the Mandelbrot Set bifurcate in the same way that they do in the Feigenbaum plot, and for good reason. Furthermore, we noticed that elsewhere along the edge of the central cardioid of the Mandelbrot Set, we can find balls that yield cycles of any magnitude, and in general the size of the ball is inversely related to the magnitude of its cycle (bigger ball = smaller cycle).

When we look at the Julia Sets we get in each of these balls, we realize that each of them have a central area, centered at the origin, and each with an infinity of “pinch points” found along its edge. The number of areas pinched together at this point exactly matches the size of the orbit contained! Given a mystery Julia Set, then, we can determine the size of its contained orbit just by its shape!

Finally, we observed that every ball connected to the central cardioid has an infinity of balls attached to it, also of decreasing size. The fractal nature of the Mandelbrot Set manifests itself here: the largest sub-ball corresponds to a bifurcation, the next largest to trifurcations, and so on. Thus, the 2-cycle found in the San Marco Julia Set can bifurcate into a 4-cycle, trifurcate into a 6-cycle, or any other multiple of two. The 3-cycle found in the Douady Rabbit can bifurcate to a 6-cycle, trifurcate to a 9-cycle, and so on for any multiple of 3. And this pattern extends to the Julia Sets themselves: the 3-ball off of the 2-ball creates a Julia Set that is a San Marco of Douady Rabbits:

San Marco made up of Douady Rabbits — The 2-cycle of the San Marco trifurcates to create a 6-cycle

And the 2-ball off of the 3-ball creates a Julia Set that is a Douady Rabbit of San Marcos:

Douady Rabbit made up of San Marcos — The 3-cycle of the Douady Rabbit bifurcates to create a different 6-cycle

With this in mind, we can play the game: Find the Julia Set. For each Julia Set in the attached file, use the shape/structure of the Julia Set to predict where in the Mandelbrot Set it can be found, and as a result the magnitude of the cycle found within it. Then verify your predictions using Fractal Zoomer.